quadrangular, pentagonal, hexagonal, &c., according as
the base has three, four, five, six sides, &c.

A sphere or globe (Fig. 53), is a solid bounded by a
uniformly curved surface, every point of which is equally
distant from the centre, a point within the sphere. A
fine passing through the centre, and terminating both
ways at the surface, is a diameter.

(Fig. 54.) (Fig. 55.) (Fig. 56.)

Cylinder. Cone, Tetrahedron.

A cylinder is a round solid of uniform thickness, of
which the ends are equal and parallel circles (Fig. 54).

A cone is a round solid, with a circle for its base, and
tapering uniformly to a point at the top (Fig. 55).

When a solid is cut through transversely by a plane
parallel to the base, the part cut off is a segment, and the
part remaining is a frustrum of the solid. The latter
term is usually limited to pyramids and cones.

(Fig. 57.) (Fig. 58.) (Fig. 59.) (Fig. 60.)

Hexahedron. Octahedron. Dodecahedron. Icosahedron.

A regidar solid is bounded by similar and regular
plane figures. There are five regular solids (Figs. 56, 57,
58, 59, 60), namely:—

The tetrahedron, bounded by four equilateral triangles.

The hexahedron, or cube, bounded by six squares.

The octahedron, bounded by eight equilateral triangles.

The dodecahedron, bounded by twelve pentagons.

The icosahedron, bounded by twenty equilateral tri-
angles.

Regular solids may be circumscribed by spheres, and
spheres may be inscribed in regular solids.

A problem is something to be done.

Geometrical construction signifies the series of opera-
tions performed with mathematical instruments, in deter-
mining the points and lines involved in the solution of a
geometrical problem.

Lines of construction are the lines used in solving
problems.

A given point; a given line. A point or line of fixed
position or length.

Describe an arc signifies that an arc of a circle is to be
drawn by the dividers, or in pencil or ink.

To set off or lay off a distance or length, is to mark on
the drawing a given distance by the dividers, the com-
passes, or the pricker ; and, with the compasses, it is done
most conveniently by describing a short arc.

Lines cut each other, when they intersect or cross.

To set off or lay out an angle is to draw a line making
a given angle with another.

To produce a line is to lengthen a given line as far as
may be necessary.

The following problems are in the first place worked
out independently of the assistance derivable from the
drawing board and squares, to embrace geometrical opera-
tions on the largest scale, in which the use of these
instruments would be inadmissible. Their use and appli-
cation will be subsequently illustrated.

SECTION I.

Problems on Straight Lines.

Problem I.—To draiv a straight line through given
points.

Let A and B be two given points, represented by the
intersection of two lines, or pricked into the surface.
Surround the points by small circles, when advisable for

Problem II.—To bisect, or divide into two equal parts,
a given straight line, or circular arc.

With a radius greater than half the given line A B,
describe arcs of circles Rom the ex-
tremities A and B, cutting each other
at the points C and D; and draw C D.
This line will cut the given line or
arc into two equal parts, at the point
E or F.

Note.—The longer the radius taken
for describing the circular arcs, the
smaller is the chance of error in the
construction, as it places the points C, D farther apart,
and facilitates the accurate drawing of the line C D in the
required position.

It is not necessary in practice to draw the complete
arcs C D. An experienced eye can readily anticipate
the points of intersection of the arcs, within narrow
limits. Nor is it necessary to do more than apply a
straight edge to these points, and to tick the point E or F.

Problem III.—To draiv a perpendicular to a straight
line.

Operate as in the foregoing problem. The line C D is
perpendicular to A B.